Question

aflati x din egalitatea:1+3+5+...+2013+x=2+4+6+...+2014

Answer 1

Raspuns:
1+3+5+......+2013+x = 2(1+2+3+.....+1007)
(1+2013)·1007/2 +x = 2·1007·1008/2
1007·1007 + x = 1007·1008
x = 1007·1008 - 1007·1007 = 1007

Answer 2

$1+3+5+...+2013+x=2+4+6+...+2014 \\\\ 2013=1+(n-1)*2 \\\\ 2013-1=(n-1)*2 \\\\ 2012=(n-1)*2 \\\\ n-1=2012:2 \\\\ n-1=1006 \\\\n=1006+1 \\\\ \boxed{n=1007} \\\\\\ \frac{2*1+(1007-1)*2}{2}*1007 \\\\ =\frac{2+1006*2}{2}*1007= \\\\ =\frac{2+2012}{2}*1007= \\\\ =\frac{2014}{2}*1007= \\\\= 1007*1007= \\\\=1007^2$

$1007^2+x=2(1+2+3+...+1007) \\\\ 1007^2+x=2* \frac{1+1007}{2}*1007 \\\\ 1007^2+x=(1+1007)*1007 \\\\ 1007^2+x=1*1007+1007*1007 \\\\ 1007^2+x=1007+1007^2 \\\\ \boxed{\boxed{x=1007}}$